In the traditional convex-concave setting (\(f(\cdot ,\delta )\)  convex
for all fixed \(\delta \in \Delta \)  and \(f(\gamma ,\cdot )\)  concave for
all fixed \(\gamma \in \Gamma \) ), the classical von Neumann minimax theorem
implies that \(\inf _{\gamma \in \Gamma }\sup _{\delta \in \Delta }f(\gamma ,\delta )=\sup _{\delta \in \Delta }\inf _{\gamma \in \Gamma }f(\gamma ,\delta )\) 
under mild conditions. In contrast to this, [1]} among
others have emphasized that in GAN applications \(f\)  is as a rule
non-convex and non-concave and the order in which minimization and
maximization are performed matters. Another point to note is that
in GAN applications the object of interest is not the optimal value
of problem (REF ), that is,
\(V_{0}=\inf _{\gamma \in \Gamma }\sup _{\delta \in \Delta }f(\gamma ,\delta ),\) 
